On Generalized Ramanujan-Style Nested Radicals for All Integers

Abstract: In this paper, we present a generalization of the celebrated nested radical expression famously attributed to Srinivasa Ramanujan, which elegantly represents the integer  through an infinite nested radical structure. We reveal the algebraic foundation underpinning this remarkable identity, rooted in the classical algebraic identity p^2-q^2=(p+q)(p-q). Extending Ramanujan’s original insight, we systematically develop a generalized form of these infinite nested radicals capable of representing explicitly and precisely any integer . Numerical illustrations via tables and examples confirm the validity and elegance of this generalized algebraic construction, significantly broadening our understanding of infinite nested radical expressions and their profound algebraic symmetries.

Keywords: Nested radical identities, Ramanujan-style radicals, Infinite nested radicals, Difference-of-squares identity, Algebraic identities.

2020 Mathematics Subject Classification: Primary: 11A99 (Number theory: miscellaneous topics); Secondary: 40A05 (Convergence and divergence of series and sequences), 97I30 (Educational exposition: Algebra).